Multi-qubit control

ABSTRACT

This disclosure relates to evaluating and improving performance of a control implementation on a quantum processor comprising multiple qubits in the presence of noise. A noise model decomposes noise interactions described by a multi-qubit noise Hamiltonian into multiple contributory noise channels. Each channel generates noise dynamics described by a unique noise-axis operator. For a given control implementation, a unique filter function represents susceptibility of the multi-qubit system to the associated noise dynamics. The filter functions are based on a frequency transformation of the noise axis operator of the corresponding noise channel to thereby evaluate the performance of the control implementation. An optimised control sequence is based on the filter function to reduce the susceptibility of the multi-qubit system to the noise channels, thereby reducing the effective interaction with the multi-qubit noise Hamiltonian. The optimised control sequence controls the quantum processor to thereby improve the performance of the control implementation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a U.S. national stage filing under 35 U.S.C.§ 371from International Application No. PCT/AU2019/050651, filed on 24 Jun.2019, and published as WO2020/019015 on 30 Jan. 2020, which claims thebenefit of priority from Australian Provisional Patent Application No2018247327 filed on 12 Oct. 2018 and Australian Patent No 2018902650filed on 23 Jul. 2018, the benefit of priority of each of which isclaimed herein, and which applications and publication are incorporatedherein by reference in their entirety.

TECHNICAL FIELD

This disclosure relates to evaluating and improving performance of acontrol implementation on a quantum processor comprising multiple qubitsin the presence of noise.

BACKGROUND

Quantum computers are an emerging technology and a number of differenttechnologies exist to implement quantum computers, including trappedions, nitrogen vacancies in diamond, superconducting circuits, andelectron and nuclear spins in semiconductors and other crystalstructures. For each of these technologies there are a wide range ofdifferent configurations and microscopic details, such as the number andspatial arrangement of qubits implemented, connectivity between qubits,available controls, or the physical environments experienced by thequbits. As a result, each quantum computer has its own specificcharacteristics that makes it different to other quantum computers.

However, one common theme for quantum computers is that quantuminformation deteriorates quickly and complex algorithms are required tocreate control protocols that can stabilize the quantum information. Dueto both the high variability between quantum computing hardware and theinherent computational challenges, it is difficult to determine andexecute these control algorithms for a particular quantum computer whichholds back the effective use of quantum computing technology.

FIG. 1 illustrates a quantum processor 101 that implements one or moreoperations on multiple qubits 102. In this example, the multiple qubitsare in the form of a 5-by-3 array of qubits but other arrangementsincluding linear arrays and other structures with many more qubits areequally possible. Similarly, nearest-neighbor couplings are illustrated,but other coupling architectures (e.g. next-nearest neighbor) are alsoequally possible. Example architectures that can be controlled using thedisclosed system include Google's Bristlecone and IBM's Q Experience.Operations implemented by quantum processor 101 may include a universalgate set to perform “circuit model” quantum computing, entanglingoperations for measurement-based quantum computing, adiabatic evolutionsfor adiabatic quantum computing, or “analog” quantum simulations.

There is now also shown a control signal source 103 which generates acontrol signal 104 to control the qubits 102. The control signal may bean RF, microwave, or optical signal depending on the physicalimplementation of the qubits. It may be delivered globally or locally tothe qubits, depending on the architecture and implementation. The signalmay be individually tailored for each qubit or may be appliedhomogeneously to all qubits. The control signal 104 may be the samesignal generally used to apply quantum operations to qubits 102, or maybe a separate parallel control system. More specifically, the controlsignal 104 may have a first component to control the qubit operations,such as load quantum information onto the qubits, perform quantum logicoperations, enable interaction, and read-out the resulting qubitinformation. On top of the first component, there is a second componentthat reduces the influence of noise on the qubits as will be describedin more detail below. The first and second components may be separate inthe sense that they are generated by separate sources or even usingseparate physical devices, such as RF, laser, MW, etc.

FIG. 1 also shows an environment 105 in which the qubits 102 operate.Environment 105 is drawn as an irregular shape to indicate that it isrelatively difficult to accurately characterize and behaves in a randommanner, though it may exhibit correlations in space and time. Moreparticularly, qubits 102 interact with the environment in an undesiredbut unavoidable way. For example, qubits 102 may interact with theenvironment 105 through various mechanisms which leads to decoherence ofthe quantum information stored on qubits 102, and decoherence-inducederrors in quantum logic operations applied to qubits 102. Environment105 may be the same across many qubits 102 in processor 101, or may varystrongly between qubits. Without countermeasures, this decoherenceresults in errors which significantly diminish the performance of theindividual quantum logic operations within the quantum processor, and ofthe ultimate algorithm executed on the quantum processor, unlessdecoherence is reduced. It is therefore the task of control signal 104to control qubits 102 such that the effects of decoherence are reduced.This control signal reduces the effects of decoherence even when controlsignal 104 itself is imperfect and contributes its owndecoherence-inducing noise characteristics.

FIG. 2 illustrates a simplified example of a single qubit 200 whosequantum state is represented as a Bloch vector on a Bloch sphere 201where the north pole 202 represents the |1> basis state (e.g. up spin)and the south pole 203 represents the |0> basis state (e.g. down spin).The current quantum information may be encoded as a superposition of thetwo basis states as indicated by first vector 203. In the currentexample the objective is to perform an idle operation, or memory. As aresult of the interaction of the qubit 200 with the environment 105 thevector rotates by an unknown and uncontrolled amount, and after acertain time arrives at a different position as indicated by secondvector 204. This corresponds to a randomization of the informationencoded in the qubit state, described by the state vector. It is nowpossible to apply a control signal that ‘flips’ the second vector 204 by180 degrees arriving at a new position as indicated by third vector 205.If the control signal is then turned off for the same time as between203 and 204, the third vector 205 will rotate exactly to the position ofthe first vector 203 due to the interaction with the environment 105. Asa result, the influence of the noise as described by the interactionwith the environment 105 in this case is perfectly eliminated. This is asimple example of open-loop (measurement-free) control which stabilizesa quantum state against environmental decoherence. This control protocoltherefore provides correction of the qubit state 203 againstenvironmental interference, but avoids direct measurement of either theenvironment 105 or the qubit states 203 or 204. This is particularlypowerful as the measurement process causes the collapse of quantuminformation, rendering it useless for quantum computation.

FIG. 3 illustrates a corresponding control sequence 300 that applies asingle control pulse 301 between t₁ 302 and t₂ 303 to flip the qubit asshown in FIG. 2. In this example, this means that RF source 103 isturned on at t₁ and turned off at t₂.

While the example in FIGS. 2 and 3 provides perfect elimination ofoutside noise due to a time-independent, or static “dephasing”interaction with environment 105, the situation is more complicated inpractical cases. In particular, the multiple qubits 102 interact witheach other and with the environment 105 in different and potentiallyrandom ways. This is especially difficult to control where only onesingle signal 104 can be applied to all qubits 102 simultaneously.

Further, most interactions with the environment 105 and between qubits102 in general cannot be corrected by a single pulse but requiremultiple complex controls forming a sequence; the timing and propertiesof each segment within the sequence is chosen to optimize theperformance of the desired operation (including but not limited tomemory (I), bit flip (X), phase flip (Z), T, S, CNOT, CPHASE, or otherentangling gates) under the influence of general time-varying noise.

It is even more complicated to determine such control sequences incircumstances where the effect of the environment 105 and controlinduced by system 104 do not quantum mechanically commute. This is aparticular condition which occurs in many quantum processors and alsoadds significant complexity to the problem of defining controls in realcircumstances. For instance, the application of a quantum bit flip (X)gate in the presence of dephasing noise constitutes such an example.Overall, it is difficult to accurately determine sequences to compensatefor noise in such a scenario.

Any discussion of documents, acts, materials, devices, articles or thelike which has been included in the present specification is not to betaken as an admission that any or all of these matters form part of theprior art base or were common general knowledge in the field relevant tothe present disclosure as it existed before the priority date of eachclaim of this application.

Throughout this specification the word “comprise”, or variations such as“comprises” or “comprising”, will be understood to imply the inclusionof a stated element, integer or step, or group of elements, integers orsteps, but not the exclusion of any other element, integer or step, orgroup of elements, integers or steps.

SUMMARY

There is provided a method for evaluating and improving performance of acontrol implementation on a quantum processor comprising multiple qubitsin the presence of noise. The method comprises:

modelling the noise by decomposing noise interactions described by amulti-qubit noise Hamiltonian into multiple contributory noise channels,each channel generating noise dynamics described by a unique noise-axisoperator A{circumflex over ( )}{(i)}, where (i) indexes the ith noisechannel;

determining, for a given control implementation, the unique filterfunction for each noise channel representing susceptibility of themulti-qubit system to the associated noise dynamics, each of themultiple filter functions being based on a frequency transformation($\mathcal{F}$) of the noise axis operator (A{circumflex over ( )}{(i)})of the corresponding noise channel (i) to thereby evaluate theperformance of the control implementation;

determining an optimised control sequence based on the filter functionto reduce the susceptibility of the multi-qubit system to the noisechannels, thereby reducing the effective interaction with themulti-qubit noise Hamiltonian;

applying the optimised control sequence to the multi-qubit system tocontrol the quantum processor to thereby improve the performance of thecontrol implementation.

It is an advantage that the method reduces the noise influence on themultiple qubits—which left unaddressed causes hardware failure—becausethe method calculates the filter function and determines anoise-suppressing control sequence based on the filter function.Therefore, the multiple qubits can be used at a higher fidelity, thatis, at a lower error rate.

The multi-qubit noise axis operators may be linear operators.

The combination of the noise axis operators may comprises a weighting ofeach noise axis operator by a random variable.

The combination may be a linear combination of the noise axis operatorsweighted by respective random variables.

Each of the multiple filter functions may be based on a frequencytransformation of a control Hamiltonian (U_c) applied to the noise axisoperator.

The control Hamiltonian may represent an operation on multiple qubits.

A gate set of the quantum processor may comprise entangling operationsand the control Hamiltonian includes the entangling operations betweenthe multiple qubits.

The control Hamiltonian (U_c) applied to the noise axis operator mayform a toggling frame noise axis operator.

The toggling frame noise axis operator may represent control dynamicsrelative to the noise axis excluding stochastic content of the noise.

The noise axis operator may be in the Hilbert space of the multiplequbits.

The noise axis operator may be time varying.

The dimensions of the noise axis operator may be equal to the dimensionsof a Hamiltonian of the multiple qubits.

The noise axis operator may be based on a non-Markovian error model.

The noise axis operator may be based on one or more measurements of anenvironment of the quantum processor.

Each filter function may be based on a sum over dimensions of the noiseaxis operator of the frequency transformation.

Determining a control sequence based on the filter function may be toreduce the noise influence on the operation on the multiple qubits.

The method may further comprise mapping operators for a multi-qubitsystem to an effective control Hamiltonian of reduced dimensionality,wherein each of the multiple filter functions is based on a frequencytransformation of the effective control Hamiltonian (U_c) applied to thenoise axis operator such that the optimized control sequence representsa multi-qubit control solution.

A quantum processor comprises multiple qubits and a controllerconfigured to:

model the noise by decomposing noise interactions described by amulti-qubit noise Hamiltonian into multiple contributory noise channels,each channel generating noise dynamics described by a unique noise-axisoperator A{circumflex over ( )}{(i)}, where (i) indexes the ith noisechannel;

determine, for a given control implementation, the unique filterfunction for each noise channel representing susceptibility of themulti-qubit system to the associated noise dynamics, each of themultiple filter functions being based on a frequency transformation($\mathcal{F}$) of the noise axis operator (A{circumflex over ( )}{(i)})of the corresponding noise channel (i) to thereby evaluate theperformance of the control implementation;

determine an optimised control sequence based on the filter function toreduce the susceptibility of the multi-qubit system to the noisechannels, thereby reducing the effective interaction with themulti-qubit noise Hamiltonian;

apply the optimised control sequence to the multi-qubit system tocontrol the quantum processor to thereby improve the performance of thecontrol implementation.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates a quantum processor.

FIG. 2 illustrates a simplified example of a single qubit whose quantumstate is represented as a Bloch vector on a Bloch sphere.

FIG. 3 illustrates a control sequence.

FIG. 4 illustrates a method for evaluating and improving performance ofa control implementation on a quantum processor.

FIG. 5 illustrates filter functions for a two-qubit system.

FIG. 6 illustrates filter functions for a two-qubit system.

DESCRIPTION OF EMBODIMENTS

As set out above, the influence of noise is a significant problem thatcauses deterioration of quantum states. In particular, for multiplequbit systems, this noise interaction is difficult to eliminate becausethe modelling and the calculation of a suitable control sequenceappropriate for either multiple interacting qubits or entanglingoperations is difficult.

This disclosure provides a method for quantum control that evaluates andimproves the performance of a quantum processor under a given controlimplementation. This means that the quantum processor is operated underthe control of a signal that essentially implements the desiredoperations, such as loading quantum information onto the multiplequbits, evolving and manipulating the quantum states for a certain time,and reading out the resulting quantum information from the qubitspotentially employing quantum error correction. These steps (andpotentially others) are referred to as control implementation as theygovern the control of the nominal behavior of the quantum processorwithout noise.

The disclosed method uses the notion of multiple noise channels which,when added together, represent the full noise influence noting that eachnoise channel may apply to all qubits so the number of qubits and thenumber of noise channels are different in most cases. The disclosedmethod then models the noise as a decomposed noise Hamiltonian and eachelement of that decomposition represents the influence of one of thenoise channels. More particularly, each element comprises a noise axisoperator that is a linear (matrix) operator and each noise axis operatorapplies to all qubits and one noise channel. When applied to the controlHamiltonian (representing the control implementation described above)the result is referred to as a “toggling frame noise operator”.

The advantage of the linear noise axis operator and the additive noisechannels is that a Fourier transform of the operator can be calculatedto determine a filter function representing the noise influence duringthe selected control. Once the filter function is known, it can be usedto improve the performance of the quantum processor by eliminating thenoise influence as much as possible.

The method is now explained on relatively broad terms with reference toFIG. 4 and after that, a rigorous mathematical formulation is provided.The broad explanation provides equation symbols that are used in themathematical formulation to establish a link between both parts of thisdisclosure.

FIG. 4 illustrates a method 400 for evaluating and improving performanceof a control implementation on a quantum processor comprising multiplequbits in the presence of noise. It is noted that method 400 may beperformed by a classical computer system comprising a processor andnon-transitory computer readable medium acting as program memory withcomputer instructions stored thereon. The processor may be acommercially available processor, such as Intel Pentium, or an FPGA orcustom made ASIC. The instructions cause the processor to perform method400. The mathematical formulations of further features of method 400 andother methods may also be implemented as program code and installed onthe program memory of the classical computer system.

The method commences by modelling 401 the noise by decomposing noiseinteractions described by a multi-qubit noise Hamiltonian H₀(t) intomultiple contributory noise channels. Each channel generates noisedynamics described by a unique noise-axis operator A^((i)) where (i)indexes the ith noise channel (Eq. 6 below);

The next step is to determine 402, for a given control implementationH_(c)(t), the unique filter function F according to

${F_{i}(\omega)} = {\frac{1}{D}{\sum\limits_{l = 1}^{D}\;{\sum\limits_{q = 1}^{D}\;{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{lq}}^{2}}}}$(which is Eq. (85) below). That is, the filter function F is calculatedfor each noise channel i and represents susceptibility of themulti-qubit system to the associated noise dynamics. Each of themultiple filter functions F is based on a frequency transformation

of the noise axis operator Ã₀ ^((i)) of the corresponding noise channeli. It is noted that Ã₀ ^((i)) in the above equation is the togglingframe noise axis operator. Determining a filter function essentiallymeans that the performance of the quantum processor is evaluated as itshows the noise influence at various noise frequencies.

Returning back to the noise characteristics, in some examplesdecoherence in quantum systems is dominated by time-dependentnon-Markovian noise processes with long correlations, characterized bylow-frequency dominated noise power spectra [1, 2 3, 4, 5]. These mayarise either from environmental fluctuations or—in the important case ofdriven quantum systems—from noise in the control device itself such asthat implementing either single or multi-qubit operations [6]. In eithercase, the result is a reduction in the fidelity of a target controloperation, including both memory and nontrivial operations. Thesephenomena present a major challenge as quantum devices move from proofof principle demonstrations to realistic applications, where performancedemands on the quantum devices are frequently extreme. Accordingly,finding ways to control quantum systems efficiently and effectively inthe presence of noise is a central task in quantum control theory.

It is then possible to determine 403 an optimised control sequence basedon the filter function to reduce the susceptibility of the multi-qubitsystem to the noise channels. Given known noise, characterized by itspower spectral density as a function of frequency, the overallperformance of the quantum operation is determined by the overlapintegral of the noise and filter function in frequency space. A controlwhich exhibits a filter function with a small numerical value where thenoise power spectral density is largest reduces the qubit's effectiveinteraction with the multi-qubit noise Hamiltonian and therefore reducesthe overall noise influence on and degradation of that gate. This may beachieved over a broad low-frequency band, or over specific narrowfrequency ranges, or both simultaneously. Therefore once calculated, thecontrol sequence may be iteratively modified to ensure both the targetoperation is achieved and the desired filtering performance achieved.

Finally, the optimised control sequence is applied 404 to themulti-qubit system to control the quantum processor to thereby improvethe performance of the control implementation in the sense that themultiple qubits are now less susceptible to noise degradation duringeither idle (memory) or nontrivial quantum logic operations, and cantherefore perform at a significantly reduced error rate.

The remainder of this disclosure sets out the formalism for computingfilter functions for a very broad class of control and noiseHamiltonians, suitable for both analytic and numeric computation, andcovering most quantum systems currently being studied around the worldfor the purpose of quantum information processing.

At first, it is helpful to express the Hamiltonian framework for thecontrol and noise interactions, though without making furtherrestrictions as to the dimension of the system, or number ofqubits/particles, or the interactions between them. ThusH(t)=H _(c)(t)+H ₀(t)  (1)where H_(c)(t) describes perfect control of the system, e.g. via anideal external driving field, and the noise Hamiltonian H₀(t) capturesundesirable time-varying interactions with a noise process. The systemevolves under the Schrödinger equation

$\begin{matrix}{{i{\overset{.}{U}( {t,0} )}} = {{H(t)}{U( {t,0} )}}} & (2) \\{{U( {t,0} )} = {{\mathcal{T}exp}( {{- i}{\int_{0}^{t}{{H( {t'} )}{{dt}'}}}} )}} & (3)\end{matrix}$where τ denoting the time-ordering operator, and the time-evolutionoperator U(t, 0) transforms an initial state |ψ(0)> to the finalU(Σ,0)|ψ(0)> after an interaction of duration τ. In the absence of noisethe total Hamiltonian reduces to H(t)=H_(c)(t), in which case the targetevolution path under ideal control is governed by

$\begin{matrix}{{i{{\overset{.}{U}}_{c}( {t,0} )}} = {{H_{c}(t)}{U_{c}( {t,0} )}}} & (4) \\{{U( {t,0} )} = {{\mathcal{T}exp}( {{- i}{\int_{0}^{t}{{H_{c}( {t'} )}{{dt}'}}}} )}} & (5)\end{matrix}$being independent of any noise processes.

Noise processes can be modelled semi-classically in terms of stochastictime-dependent fluctuating classical noise fields. Without furtherspecificity, the noise Hamiltonian is expressed as a combination of nnoise channels, which may be written as a sum of n noise channels

$\begin{matrix}{{H_{0}(t)} = {\sum\limits_{i = 1}^{n}{{\beta_{i}(t)}{A^{(i)}(t)}}}} & (6)\end{matrix}$where the noise axis operator A^((i)) is not a stochastic variable,rather an operator defining the “axis” of the noise (possibly changingin time) in the Hilbert space of the system. In this model the noisefields A(t) are assumed to be classical random variables with zero meanand non-Markvovian power spectra. They are also assumed wide sensestationary (w.s.s.) and independent. The assumption of independence isreasonable, for instance, in the case of a driving field where randomfluctuations in frequency and amplitude arise from different physicalprocesses. A general model including correlations between noiseprocesses is possible, however, following the approach outlined by Greenet al. [7]. The assumption of w.s.s. implies the autocorrelationfunctionsC _(i)(t ₂ −t ₁)≡<(β_(i)(t ₁)β_(i)(t ₂)>, i∈{1, . . . ,n}  (7)depend only on the time difference t₁−t₂. Moreover the autocorrelationfunction for each noise channel may be related to the Fourier transformof the associated PSD S_(i)(ω) using the Wiener-Khinchin Theorem [8].Namely

$\begin{matrix}{{C_{i}( {t_{2} - t_{1}} )} = {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{S_{i}(\omega)}e^{i\;{\omega{({t_{2} - t_{1}})}}}d\;{\omega.}}}}} & (8)\end{matrix}$

The assumption of independence implies the cross-correlation functionsvanish. That is,<β_(j)(t ₁)β_(k)(t ₂)>=0, j≠k∈{1, . . . ,n}  (9)

The angle brackets here denote a time average of the stochasticvariables. Finally, this model permits access to a wide range ofparameter regimes, from quasistatic (noise slow compared to H_(c)(t)) tothe limit in which the noise fluctuates on timescales comparable to orfaster than H_(c)(t).

These noise Hamiltonians generate uncontrolled rotations in the qubitdynamics, leading to errors in the evolution path (and hence the finalstate) relative to the target transformation intended under H_(c)(t). Anestimate for this error is derived in the following description using afirst order expansion of operators, resulting in a generic descriptionin terms of associated filter functions.

Generic Filter Functions

The proposed process uses a measure for the operational fidelity in thepresence of both noise and the relevant control. For this, the methoddeveloped by Green et al. [18] can be referenced. We note that fidelitymay not be the only relevant metric—a wide range of measurableparameters may be calculated in the same framework. In the currentframework the error contributed by the noise fields over the duration ofthe control is approximated, to first order, via a truncated Magnusexpansion. Each noise field then contributes a term to the gateinfidelity in the spectral domain expressed as an overlap integralbetween the noise power spectrum and an appropriate generalizedfilter-transfer function.

Operational Fidelity

In the absence of noise interactions, the target state evolution isdescribed by U_(c)(t) and determined by Eq (4). Noise interactions underH₀ steer the operation away from this target resulting in the netoperation U(t) determined by Eq (18). Average operational fidelity maytherefore be expressed

$\begin{matrix}{{\mathcal{F}_{av}(\tau)} = \langle {{\frac{1}{D}{{Tr}( {{U_{c}^{\dagger}(\tau)}{U(\tau)}} )}}}^{2} \rangle} & (10)\end{matrix}$effectively measuring the extent to which the intended and realizedoperators “overlap”, as captured by the Hilbert-Schmidt inner product[9]. Here D is the dimension of the Hilbert space. This ensures that werecover unit fidelity when there is no noise. Namely, if U=U_(c) weobtain

$\begin{matrix}{{\mathcal{F}_{av}(\tau)} = {\langle {{\frac{1}{D}{{Tr}( {U_{c}^{\dagger}U_{c}} )}}}^{2} \rangle = {\langle {{\frac{1}{D}{{Tr}( \overset{\hat{}}{\mathbb{I}} )}}}^{2} \rangle = {\langle {{\frac{1}{D}D}}^{2} \rangle = 1}}}} & (11)\end{matrix}$Toggling Frame

Computing the evolution dynamics such to evaluate Eq. (10) ischallenging since the control and noise Hamiltonians may not commute atdifferent times; sequential application of the resulting time-dependent,non-commuting operations gives rise to both dephasing and depolarizationerrors, mandating approximation methods.

The proposed error model assumes non-dissipative qubit evolution withboth control and noise interactions resulting in unitary rotations.Hence, the evolution operator is approximated as a unitary using aMagnus expansion [10, 11]. This involves moving to a frame co-rotatingwith the control known as the toggling frame, see also [12]. Thisapproach allows to separate the part of the system evolution due solelyto the control from the part affected by environmental coupling.

Let the error between the full and ideal unitaries, U and U_(c), beexpressed in terms of the error unitary Ũ such that U=U_(c)Ũ. In thiscase we obtain the following equivalent relationsU=U _(c) Ũ  (12)Ũ=U _(c) ^(†) U  (13)U _(c) ^(†) =ŨU ^(†).  (14)

Computation of Eq. (10) may therefore be performed using Eq. (13),yielding

$\begin{matrix}{{\mathcal{F}_{av}(\tau)} = \langle {{\frac{1}{D}{{Tr}( {\overset{\sim}{U}(\tau)} )}}}^{2} \rangle} & (15)\end{matrix}$

Perfect control therefore corresponds to Ũ(τ)→{tilde over (∥)}. But howto compute Ũ(τ) ? Taking the time derivative of Eq. (13) we obtain{dot over (Ũ)}={dot over (U)} _(c) ^(†) U+U _(c) ^(†) {dot over(U)}  (16)={dot over (U)} _(c) ^(†)(U _(c) Ũ)+(ŨU ^(†)){dot over (U)}  (17)

Now from Eq. (18) and Eq. (4), and observing that H_(c) ^(†)=H_(c), wealso havei{dot over (U)}=HU

{dot over (U)}=−iHU  (18)i{dot over (U)}=H _(c) U _(c)

{dot over (U)} _(c) ^(†) =iU _(c) ^(†) H _(c)  (19)

Substituting these into Eq. (17) we therefore have{dot over (Ũ)}=iU _(c) ^(†) H _(c)(U _(c) Ũ)−i(ŨU ^(†))HU  (20)=i(U _(c) ^(†) H _(c) U _(c))Ũ−iŨ(U ^(†) HU)  (21)=i(U _(c) ^(†) H _(c) U _(c))Ũ−i(ŨU ^(†))HU  (22)

Now substituting in Eq. (12) and Eq. (14) yields{dot over (Ũ)}=i(U _(c) ^(†) H _(c) U _(c))Ũ−iU _(c) ^(†) HU _(c)Ũ  (23)=i(U _(c) ^(†) H _(c) U _(c) −U _(c) ^(†) HU _(c))Ũ  (24)=i(U _(c) ^(†) H _(c) U _(c) −U _(c) ^(†)(H _(c) +H ₀)U _(c))Ũ  (25)=−i(U _(c) ^(†) H ₀ U _(c))Ũ  (26)

Thus, defining the toggling frame Hamiltonian{tilde over (H)} ₀(t)≡U _(c) ^(†)(t)H ₀(t)U _(c)(t)  (27)the error propagator is Ũ satisfies the Schrödinger equationi{dot over (Ũ)}(t)={tilde over (H)} ₀(t)Ũ(t).  (28)and we may perform a Magnus expansion in this frame to approximate Ũ(τ).Magnus Expansion

Assuming convergence of the Magnus series [11] it is possible to obtainan arbitrarily accurate unitary estimate of the error propagator Ũgoverned by Eq. (28), by performing a Magnus expansion. In particular,at the end of the evolution time τ, we may writeŨ(τ)=exp[−iΦ(τ)]  (29)where the effective error operator is expands as

$\begin{matrix}{{\Phi(\tau)} = {\sum\limits_{\mu = 1}^{\infty}{\Phi_{\mu}(\tau)}}} & (30)\end{matrix}$with the first few Magnus expansion terms expressed as

$\begin{matrix}\begin{matrix}{{\Phi_{1}(\tau)} = {\int_{0}^{\tau}{{dt}{{\overset{\sim}{H}}_{0}(t)}}}} \\{{\Phi_{2}(\tau)} = {{- \frac{i}{2}}{\int_{0}^{\tau}{dt_{1}{\int_{0}^{1}{d{t_{2}\lbrack {{{\overset{\sim}{H}}_{0}( t_{1} )},{{\overset{\sim}{H}}_{0}( t_{2} )}} \rbrack}}}}}}} \\{{\Phi_{3}(\tau)} = {\frac{1}{6}{\int_{0}^{\tau}{dt_{1}{\int_{0}^{t_{1}}{dt_{2}{\int_{0}^{t_{2}}{dt_{3}\{ {\lbrack {{{\overset{\sim}{H}}_{0}( t_{1} )},\lbrack {{{\overset{\sim}{H}}_{0}( t_{2} )},{{\overset{\sim}{H}}_{0}( t_{3} )}} \rbrack} \rbrack +} }}}}}}}} \\ \lbrack {{{\overset{\sim}{H}}_{0}( t_{3} )},\lbrack {{{\overset{\sim}{H}}_{0}( t_{2} )},{{\overset{\sim}{H}}_{0}( t_{1} )}} \rbrack} \rbrack \} \\{\ldots}\end{matrix} & (31)\end{matrix}$

These generally take the form of time-ordered integrals over nestedcommutators in {tilde over (H)}₀(t). Assuming the noise fields β_(i)(t)are sufficiently weak, we may make the lowest order approximation thatŨ(τ)≈exp[−iΦ ₁(τ)]  (32)by truncating the error operator to first order as

$\begin{matrix}{{{\Phi(\tau)} \approx {\Phi_{1}(\tau)}} = {\int_{0}^{\tau}{{dt}{{\overset{\sim}{H}}_{0}(t)}}}} & (33)\end{matrix}$Exponential Expansion of First Order Error Unitary

We assert the toggling frame Hamiltonian is both Hermitian andtraceless:{tilde over (H)} ₀ ={tilde over (H)} ₀ ^(†) , Tr({tilde over (H)}₀)=0.  (34)From Eq. (33), and due to the linearity of integration, it follows thefirst order error propagator also satisfies these properties:Φ₁=Φ₁ ^(\) , Tr(Φ₁)=0.  (35)

Now, for an n×n real or complex matrix X, the exponentiated matrix maybe expressed as the power series

$\begin{matrix}{{\exp\lbrack X\rbrack} = {\sum\limits_{k = 0}^{\infty}{\frac{1}{k!}{X^{k}.}}}} & (36)\end{matrix}$

Consequently, Eq. (32) expands as

$\begin{matrix}{{\overset{\sim}{U}(\tau)} = {\overset{\hat{}}{\mathbb{I}} - {i\Phi_{1}} - {\frac{1}{2}\Phi_{1}^{2}} + \ldots}} & (37) \\{= {\overset{\hat{}}{\mathbb{I}} - {i\Phi_{1}} - {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} + \ldots}} & (38)\end{matrix}$where we have truncated to the lowest order term that (non-trivially)survives the trace operation in Eq. (15).Operational infidelity: first order approximation

Substitution of this expansion into Eq. (15) yields

$\begin{matrix}{{\mathcal{F}_{av}(\tau)} = \langle {{\frac{1}{D}{{Tr}( {\overset{\hat{}}{\mathbb{I}} - {i\;\Phi_{1}} - {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} + \ldots} )}}}^{2} \rangle} & (39) \\{\approx \langle {{{\frac{1}{D}{{Tr}( \overset{\hat{}}{\mathbb{I}} )}} - {\frac{1}{D}{{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}}}}^{2} \rangle} & (40) \\{= \langle {{1 - {\frac{1}{D}{{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}}}}^{2} \rangle} & (41) \\{= \langle {\lbrack {1 - {\frac{1}{D}{{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}}} \rbrack^{*}\lbrack {1 - {\frac{1}{D}{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}} \rbrack} \rangle} & (42) \\{= \langle {1 - {\frac{2}{D}{{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}} + {\mathcal{O}( {\Phi_{1}}^{4} )}} \rangle} & (43)\end{matrix}$

The last line uses the following result. For the n×n Hermitian operatorA=A ^(†)

A _(ij)=[A ^(\)]_(ij) =A* _(ji)  (44)therefore

$\begin{matrix}{\lbrack {AA^{\dagger}} \rbrack_{ij} = {{\sum\limits_{k = 1}^{n}{A_{ik}\lbrack A^{\dagger} \rbrack}_{kj}} = {\sum\limits_{k = 1}^{n}{A_{ik}A_{jk}^{*}}}}} & (45)\end{matrix}$therefore

$\begin{matrix}{{{Tr}\lbrack {AA^{\dagger}} \rbrack} = {{\sum\limits_{i = 1}^{n}\lbrack {AA^{\dagger}} \rbrack_{ii}} = {{\sum\limits_{i = 1}^{n}{\sum\limits_{k = 1}^{n}{A_{ik}A_{ik}^{*}}}} = {{\sum\limits_{i = 1}^{n}{\sum\limits_{k = 1}^{n}{A_{ik}}^{2}}} \in {\mathbb{R}}}}}} & (46)\end{matrix}$

Consequently

$\begin{matrix}{{{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}^{*} = {T{r( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}}} & (47)\end{matrix}$

Ignoring terms beyond

(|Ψ₁|²) we therefore obtain

$\begin{matrix}{{\mathcal{F}_{av}(\tau)} = \langle {1 - {\frac{2}{D}{{Tr}( {\frac{1}{2}\Phi_{1}\Phi_{1}^{\dagger}} )}}} \rangle} & (48) \\{= {1 - {\frac{2}{2D}\langle {{Tr}( {\Phi_{1}\Phi_{1}^{\dagger}} )} \rangle}}} & (49) \\{= {1 - {\frac{1}{D}{{Tr}( \langle {\Phi_{1}\Phi_{1}^{T}} \rangle )}}}} & (50)\end{matrix}$Convert to Frequency Domain

Therefore, a first order approximation to infidelity can be expressed as

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\frac{1}{D}{{Tr}( \langle {\Phi_{1}\Phi_{1}^{\dagger}} \rangle )}}} & (51) \\{= {\frac{1}{D}{{Tr}( \langle {( {\int_{0}^{\tau}{{dt}_{1}{{\overset{\sim}{H}}_{0}( t_{1} )}}} )( {\int_{0}^{\tau}{{dt}_{2}{{\overset{\sim}{H}}_{0}( t_{2} )}}} )^{\dagger}} \rangle )}}} & (52) \\{= {\frac{1}{D}{{Tr}( \langle {\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{dt}_{2}{{\overset{\sim}{H}}_{0}( t_{1} )}{\overset{\sim}{H}( t_{2} )}^{\dagger}}}}} \rangle )}}} & (53)\end{matrix}$

Now, from the definitions in Eq. (6) and (27) and we have

$\begin{matrix}{{\overset{\sim}{H}}_{0} = {{U_{c}^{\dagger}H_{0}U_{c}} = {{{U_{c}^{\dagger}( {\sum\limits_{i = 1}^{n}\;{\beta_{i}A^{(i)}}} )}U_{c}} = {{\sum\limits_{i = 1}^{n}\;{\beta_{i}U_{c}^{\dagger}A^{(i)}U_{c}}} \equiv {\sum\limits_{i = 1}^{n}\;{\beta_{i}{\overset{\sim}{A}}_{0}^{(i)}}}}}}} & (56)\end{matrix}$where we have defined the toggling frame noise axis operatorÃ ₀ ^((i)) ≡U _(c) ^(\) A ^((i)) U _(c).  (57)which holds for arbitrary Hilbert space dimension D.

NOTE: the noise axis operator A^((i)) is not a stochastic variable,rather an operator defining the axis of the noise. Therefore thetoggling frame noise axis operator Ã₀ ^((i)), consisting of atransformed operator via conjugation by the control unitary U_(c), doesnot capture any information about the stochastic content of the noise,rather it captures information about the control dynamics relative to anoise axis of interest.

Inserting this into Eq. (6), we therefore have

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\frac{1}{D}{{Tr}( \langle {\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{{dt}_{2}( {\sum\limits_{i = 1}^{n}\;{{\beta_{i}( t_{1} )}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}}} )}( {\sum\limits_{j = 1}^{n}\;{{\beta_{j}( t_{2} )}{{\overset{\sim}{A}}_{0}^{(j)}( t_{2} )}}} )^{\dagger}}}}} \rangle )}}} & (58) \\{\mspace{40mu}{= {\frac{1}{D}{{Tr}( \langle {\sum\limits_{i = 1}^{n}\;{\sum\limits_{j = 1}^{n}\;{\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{dt}_{2}{\beta_{i}( t_{1} )}{\beta_{j}( t_{2} )}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}( {{\overset{\sim}{A}}_{0}^{(j)}( t_{2} )} )^{\dagger}}}}}}} \rangle )}}}} & (59) \\{\mspace{40mu}{= {\frac{1}{D}{{Tr}( {\sum\limits_{i = 1}^{n}\;{\sum\limits_{j = 1}^{n}\;{\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{dt}_{2}\langle {{\beta_{i}( t_{1} )}{\beta_{j}( t_{2} )}{\overset{\sim}{\rangle A}}_{0}^{(i)}( t_{1} )( {{\overset{\sim}{A}}_{0}^{(j)}( t_{2} )} )^{\dagger}} \rangle}}}}}} )}}}} & (60)\end{matrix}$

From Eq. (9) all cross correlations are zero, consequently

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\frac{1}{D}{{Tr}( {\sum\limits_{i = 1}^{n}\;{\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{dt}_{2}\langle {{\beta_{i}( t_{1} )}{\beta_{i}( t_{2} )}} \rangle{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}( {{\overset{\sim}{A}}_{0}^{(i)}( t_{2} )} )^{\dagger}}}}}} )}}} & (61) \\{= {\frac{1}{D}{{Tr}( {\sum\limits_{i = 1}^{n}\;{\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{dt}_{2}{C_{i}( {t_{2} - t_{1}} )}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}( {{\overset{\sim}{A}}_{0}^{(i)}( t_{2} )} )^{\dagger}}}}}} )}}} & (62)\end{matrix}$

We may now move to the frequency domain using the Wienner-Khintchinetheorem described in Eq. (8). Namely

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\frac{1}{D}{{Tr}( {\sum\limits_{i = 1}^{n}\;{\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{{dt}_{2}( {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{S_{i}(\omega)}e^{{i\omega}{({t_{2} - t_{1}})}}d\;\omega}}} )}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}( {{\overset{\sim}{A}}_{0}^{(i)}( t_{2} )} )^{\dagger}}}}}} )}}} & (63) \\{\;{= {\sum\limits_{i = 1}^{n}\;{\frac{1}{D}\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{{d\omega S}_{i}(\omega)}{{Tr}( {\int_{0}^{\tau}{{dt}_{1}{\int_{0}^{\tau}{{dt}_{2}e^{i\;\omega\; t_{2}}e^{{- i}\;\omega\; t_{1}}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}( {{\overset{\sim}{A}}_{0}^{(i)}( t_{2} )} )^{\dagger}}}}} )}}}}}}} & (64) \\{\mspace{11mu}{= {\sum\limits_{i = 1}^{n}\;{\frac{1}{D}\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{{d\omega S}_{i}(\omega)}{{Tr}( {\lbrack {\int_{0}^{\tau}{{dt}_{1}e^{{- {i\omega}}\; t_{1}}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}}} \rbrack\lbrack {\int_{0}^{\tau}{{dt}_{2}{e^{{i\omega t}_{2}}( {{\overset{\sim}{A}}_{0}^{(i)}( t_{2} )} )}^{\dagger}}} \rbrack} )}}}}}}} & (65) \\{\mspace{14mu}{= {\sum\limits_{i = 1}^{n}\;{\frac{1}{D}\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{{d\omega S}_{i}(\omega)}{{Tr}( {\lbrack {\int_{0}^{\tau}{{dt}_{1}e^{{- i}\;\omega\; t_{1}}{{\overset{\sim}{A}}_{0}^{(i)}( t_{1} )}}} \rbrack\lbrack {\int_{0}^{\tau}{{dt}_{2}{e^{- {i\omega t}_{2}}( {{\overset{\sim}{A}}_{0}^{(i)}( t_{2} )} )}^{\dagger}}} \rbrack} )}}}}}}} & (66)\end{matrix}$

Defining the Fourier transform of the operators Ã₀ ^((i))(t) as

$\begin{matrix}{{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} = {\int_{- \infty}^{\infty}{{dte}^{- {i\omega t}}{{\overset{\sim}{A}}_{0}^{(i)}(t)}}}} & (67)\end{matrix}$we therefore obtain

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\sum\limits_{i = 1}^{n}\;{\frac{1}{2\pi}{\int_{- \infty}^{\infty}{d\;\omega\;{S_{i}(\omega)}\frac{1}{D}{{Tr}( {{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger}} )}}}}}} & (68)\end{matrix}$

Thus we may finally express the infidelity as

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\sum\limits_{i = 1}^{n}\;{\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{S_{i}(\omega)}{F_{i}(\omega)}{d\omega}}}}}} & (69)\end{matrix}$where the filter functions F_(i)(ω) associated with each noise channelis defined by

$\begin{matrix}{{F_{i}(\omega)} \equiv {\frac{1}{D}{{{Tr}( {{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger}} )}.}}} & (70)\end{matrix}$

NOTE: the toggling frame noise axis operator is the control-transformedoperator defining the noise axis, and therefore the Fourier transformsin Eq. (70) capture information about the control, relative to the noisechannel of interest, but do not depend on any stochastic noisevariables.

SinceA ^((i))=(Ã ^((i)))^(†)  (71)it follows(Ã ₀ ^((i)))^(\)=(U _(c) ^(\) A ^((i)) U _(c))^(†)=(U _(c))^(\)(A^((i)))^(†)(U _(c) ^(†))^(†) =U _(c) ^(†) A ^((i)) U _(c) =Ã ₀^((i))  (72)

Consequently,[(Ã ₀ ^((i)))^(†)]_(jk)=[Ã ₀ ^((i))]_(kj) ^(*)  (73)

Now, applying Eq. (67) element wise, we therefore have

$\begin{matrix}{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger} \rbrack_{jk} = \lbrack {\int_{- \infty}^{\infty}{{dte}^{- {i\omega t}}{{\overset{\sim}{A}}_{0}^{(i)}(t)}}} \rbrack_{jk}^{\dagger}} & (74) \\{= \lbrack {\int_{- \infty}^{\infty}{{dte}^{+ {i\omega t}}( {{\overset{\sim}{A}}_{0}^{(i)}(t)} )}^{\dagger}} \rbrack_{jk}} & (75) \\{= \lbrack {\int_{- \infty}^{\infty}{{dte}^{+ {i\omega t}}( {{\overset{\sim}{A}}_{0}^{(i)}(t)} )}^{\dagger}} \rbrack_{jk}} & (76) \\{= {\int_{- \infty}^{\infty}{{dte}^{+ {i\omega t}}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}_{jk}^{*}}} & (77) \\{= ( {\int_{- \infty}^{\infty}{{dte}^{- {i\omega t}}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}_{kj}} )^{*}} & (78) \\{= \lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{kj}^{*}} & (79)\end{matrix}$

Therefore in (70)

$\begin{matrix}{\lbrack {{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger}} \rbrack_{jk} = {\sum\limits_{q = 1}^{D}\;{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{jq}\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger} \rbrack}_{qk}}} & (80) \\{\mspace{205mu}{= {\sum\limits_{q = 1}^{D}\;{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{jq}\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack}_{kq}^{*}}}} & (81)\end{matrix}$

Therefore

$\begin{matrix}{{{Tr}\lbrack {{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger}} \rbrack} = {\sum\limits_{l = 1}^{D}\;\lbrack {{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}{\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack}^{\dagger}} \rbrack_{ll}}} & (82) \\{= {\sum\limits_{l = 1}^{D}\;{\sum\limits_{q = 1}^{D}\;{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{lq}\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack}_{lq}^{*}}}} & (83) \\{= {\sum\limits_{l = 1}^{D}\;{\sum\limits_{q = 1}^{D}\;{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{lq}}^{2}}}} & (84)\end{matrix}$

The filter function associated with the i th noise channel thereforefinally takes the form

$\begin{matrix}{{F_{i}(\omega)} = {\frac{1}{D}{\sum\limits_{l = 1}^{D}\;{\sum\limits_{q = 1}^{D}\;{\lbrack {\mathcal{F}\lbrack {\overset{\sim}{A}}_{0}^{(i)} \rbrack} \rbrack_{lq}}^{2}}}}} & (85)\end{matrix}$

SUMMARY

We may therefore summarize as follows. For a quantum system of dimensionD, let the Hamiltonian be writtenH(t)=H _(c)(t)+H ₀(t)  (86)where H_(c)(t) describes perfect control, satisfyingi{dot over (U)} _(c)(t,0)=H _(c)(t)U _(c)(t,0),  (87)and the H₀(t) captures the interaction with n independent noisechannels, each modelled semi-classically as

$\begin{matrix}{{H_{0}(t)} = {\sum\limits_{i = 1}^{n}\;{{\beta_{i}(t)}{A^{(i)}(t)}}}} & (88)\end{matrix}$where the noise fields β_(i)(t) are assumed to be a classical zero-meanw.s.s. processes with associated PSD S_(i)(ω). Now, defining thetoggling frame noise axis operatorsÃ ₀ ^((i))(t)≡U _(c) ^(†)(t)A ^((i))(t)U _(c)(t)  (89)and the associated Fourier transforms

$\begin{matrix}{{\mathcal{F}\lbrack {{\overset{\sim}{A}}_{0}^{(i)}(t)} \rbrack} \equiv {\int_{- \infty}^{\infty}{{dte}^{- {i\omega t}}{{\overset{\sim}{A}}_{0}^{(i)}(t)}}}} & (90)\end{matrix}$we may express the operational infidelity, to first order, as

$\begin{matrix}{{1 - \mathcal{F}_{av}} = {\sum\limits_{i = 1}^{n}\;{\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{S_{i}(\omega)}{F_{i}(\omega)}{d\omega}}}}}} & (91)\end{matrix}$where the filter functions F_(i)(ω) associated with each noise channelare computed as

$\begin{matrix}{{F_{i}(\omega)} = {\frac{1}{D}{\sum\limits_{l = 1}^{D}\;{\sum\limits_{q = 1}^{D}\;{{\lbrack {\mathcal{F}\lbrack {{\overset{\sim}{A}}_{0}^{(i)}(t)} \rbrack} \rbrack_{lq}}^{2}.}}}}} & (92)\end{matrix}$

That is, take the Fourier transform of each element of thetime-dependent operator Ã₀ ^((i))(t), sum the complex modulus square ofevery element, and divide through by the dimension D of the quantumsystem. This result is appropriate for a general quantum system ofHilbert-space dimension D, which may be a multilevel “qudit” (A quantumunit of information that may take any of d states, where d is avariable), or multiple interacting qubits.

Decoherence in real driven systems is predominantly due to low-frequencycorrelated noise environments. This strongly motivates this disclosureof bounded-strength control as a noise filtering problem usingtime-dependent, non-Markovian error models, especially in the context ofD-dimensional multiqubit or generic multilevel “qudit” systems.Moreover, the complications of treating bounded-strength control—due tothe continual presence of noise interactions during control operationsand the resulting nonlinear dynamics—can be addressed by the disclosedstreamlined approach to the design of noise-filtering control for Ddimensional systems.

The filter-transfer function framework described herein may take asinput experimentally measurable characteristics of theenvironment—namely noise power spectra—and provides a numerically viableframework for both control construction and the calculation of predictedoperational fidelities. It also efficiently captures the controlnonlinearities implicit in situations where control and noiseHamiltonians do not commute. This framework is applicable to quantumsystems beyond the original single-qubit systems studied, with modestassumptions about the form of noise axis operators describing therelevant noise channels in the system of interest.

The filter function presented in Eq. (85) may be implemented as a Pythonmodule for numeric computation of filter functions or in otherprogramming languages including C++ and Java. The filter function hasbeen tested against familiar cases where analytic forms are known, withgood agreement confirmed over a frequency band limited by the samplingrate of the FFT used to numerically compute the Fourier transforms.Relevant example sample rates are 1,000 samples per operation, thereforeranging from approximately 1 ks/sec to 100 Gs/sec depending on thetimescale of the relevant operation but not confined to this range.

Single-Qubit Examples

Single-Qubit Control with Single-Axis Amplitude Modulation

Consider a single qubit (D=2) system with control Hamiltonian

$\begin{matrix}{{H_{c}(t)} = {\frac{\Omega(t)}{2}{\hat{\sigma}}_{x}}} & (93)\end{matrix}$e.g. using a Slepian modulation for Ω(t) to probe amplitude-modulated{circumflex over (σ)}_(x) noise. Therefore the noise Hamiltonian takesthe form

$\begin{matrix}{{H_{0}(t)} = {{\beta_{x}(t)}\frac{\Omega(t)}{2}{{\hat{\sigma}}_{x}.}}} & (94)\end{matrix}$

The noise axis operator therefore takes the form

$\begin{matrix}{A^{(x)} = {\frac{\Omega(t)}{2}{\hat{\sigma}}_{x}}} & (95)\end{matrix}$so that the toggling frame noise axis operator is given by

$\begin{matrix}{{{{\overset{\sim}{A}}_{0}^{(x)}(t)} \equiv {{U_{c}^{\dagger}(t)}{A^{(x)}(t)}{U_{c}(t)}}} = {\frac{\Omega(t)}{2}{\hat{\sigma}}_{x}}} & (96)\end{matrix}$since the control unitaries and noise axis operators commute formultiplicative amplitude noise. Consequently

$\begin{matrix}{{\mathcal{F}\lbrack {{\overset{\sim}{A}}_{0}^{(i)}(t)} \rbrack} = {{{\mathcal{F}\lbrack \frac{\Omega(t)}{2} \rbrack}{\hat{\sigma}}_{x}} = {\frac{1}{2}\begin{bmatrix}0 & {\mathcal{F}\lbrack {\Omega(t)} \rbrack} \\{\mathcal{F}\lbrack {\Omega(t)} \rbrack} & 0\end{bmatrix}}}} & (97)\end{matrix}$

Therefore

$\begin{matrix}{{F_{i}(\omega)} = {\frac{1}{2}{\sum\limits_{l = 1}^{2}\;{\sum\limits_{q = 1}^{2}\;{\lbrack {\mathcal{F}\lbrack {{\overset{\sim}{A}}_{0}^{(i)}(t)} \rbrack} \rbrack_{lq}}^{2}}}}} & (98) \\{\mspace{56mu}{= {\frac{1}{2}( {{{\frac{1}{2}{\mathcal{F}\lbrack {\Omega(t)} \rbrack}}}^{2} + {{\frac{1}{2}{\mathcal{F}\lbrack {\Omega(t)} \rbrack}}}^{2}} )}}} & (99) \\{\mspace{50mu}{= {\frac{1}{2}( {2{{\mathcal{F}\lbrack \frac{\Omega(t)}{2} \rbrack}}^{2}} )}}} & (100) \\{\mspace{50mu}{= {{\mathcal{F}\lbrack \frac{\Omega(t)}{2} \rbrack}}^{2}}} & (101)\end{matrix}$

That is, the modulus square of the Fourier transform of the amplitudemodulated envelope (e.g. a Slepian function).

Single-Qubit Dynamic Decoupling

Dynamic decoupling (DD) fits into the above description by requiring thecontrol Hamiltonian to take the form of a series of instantaneous (“bangbang”) operations. This can be phrased as a form of amplitudemodulation, switching between values of zero (off) and infinite (on).The canonical DD scheme consists of a series n instantaneous π pulsesaround {circumflex over (σ)}_(x), executed at times {t_(j)}^(j=1) _(n)over the time interval [0, τ], where we define t₀=0 and t_(n+1)=τ. Weintroduce the fractional time δ_(j)∈[0,1] defined by t_(j)/τ denotingthe normalized pulse locations. In this case the sequence(δ₁,δ₂, . . . ,δ_(n))τ  (102)completely determines the DD structure and therefore its properties.Therefore the control Hamiltonian (D=2) takes the form

$\begin{matrix}{{H_{c}(t)} = {\frac{\Omega(t)}{2}{\hat{\sigma}}_{x}}} & (103)\end{matrix}$where the amplitude modulation envelope Q(t) takes values 0 or Ω,defining a train of n pulses centred at times t_(j), each pulse havingan area π and duration τ_(j)=π/Ω. In the bang-bang limit, τ_(j)→0 andΩ→∞. The control amplitude envelope may then be written as a Dirac comb:

$\begin{matrix}{{\Omega(t)} = {\pi{\sum\limits_{j = 1}^{n}\;{{\delta( {t - t_{j}} )}{{\hat{\sigma}}_{x}.}}}}} & (104)\end{matrix}$

Therefore the control unitary takes the form

$\begin{matrix}{{U_{c}(t)} = {\exp\lbrack {{- {i( {\frac{\pi}{2}{\int_{0}^{t}{{{dt}'}{\sum\limits_{j = 1}^{n}\;{\delta( {{t'} - t_{j}} )}}}}} )}}{\hat{\sigma}}_{x}} \rbrack}} & (105) \\{\mspace{50mu}{= {\exp\lbrack {{- i}{\sum\limits_{j = 1}^{n}\;{( {\frac{\pi}{2}{\int_{0}^{t}{{{dt}'}{\delta( {{t'} - t_{j}} )}}}} ){\hat{\sigma}}_{x}}}} \rbrack}}} & (106) \\{\mspace{45mu}{= {\exp\lbrack {{- i}\frac{\pi}{2}{\sum\limits_{j = 1}^{n}\;{{\Theta_{j}(t)}{\hat{\sigma}}_{x}}}} \rbrack}}} & (107)\end{matrix}$where the jth integral evaluates as

$\begin{matrix}{{\int_{0}^{t}{{{dt}'}{\delta( {{t'} - t_{j}} )}}} = {\Theta_{j}(t)}} & (108)\end{matrix}$and Θ(t) is the Heaveside function defined as

$\begin{matrix}{{\Theta_{j}(t)} \equiv \{ \begin{matrix}0 & {t < t_{j}} \\1 & {t \geq t_{j}}\end{matrix} } & (109)\end{matrix}$

Therefore

$\begin{matrix}{{{S(t)} \equiv {\sum\limits_{j = 1}^{n}\;{\Theta_{j}(t)}}} = \{ \begin{matrix}0 & {t_{0} \leq t < t} \\1 & {t_{1} \leq t < t_{2}} \\\cdots & \cdots \\j & {t_{j} \leq t < t_{j + 1}} \\\cdots & \cdots \\n & {t_{n} \leq t < t_{n + 1}}\end{matrix} } & (110)\end{matrix}$where S(t) counts the number of pulses that have been executed by timet.

Therefore,

$\begin{matrix}{{U_{c}(t)} = {{\exp\lbrack {{- i}\frac{\pi}{2}{S(t)}{\hat{\sigma}}_{x}} \rbrack} = \{ \begin{matrix}\hat{II} & {{for}\mspace{14mu}{S(t)}\mspace{14mu}{even}} \\{{- i}{\hat{\sigma}}_{x}} & {{for}\mspace{14mu}{S(t)}\mspace{14mu}{odd}}\end{matrix} }} & (111)\end{matrix}$

In particular

$\begin{matrix}{{U_{c}(t)} = \{ \begin{matrix}\hat{\mathbb{I}} & {t_{0} \leq t < t_{1}} & {{S(t)} = 0} \\{\hat{\sigma}}_{x} & {t_{1} \leq t < t_{2}} & {{S(t)} = 1} \\\hat{\mathbb{I}} & {t_{2} \leq t < t_{3}} & {{S(t)} = 2} \\{\hat{\sigma}}_{x} & {t_{3} \leq t < t_{4}} & {{S(t)} = 3} \\\ldots & \ldots & \;\end{matrix} } & (112)\end{matrix}$where the complex factor −i in front of {circumflex over (σ)}_(x) hasbeen ignored as this will only contributes a global phase. We may writethis more compactly by introducing the square-wave (switching) functiony(t), which takes values ±1, starting with y(0)=−1, such that the jthsign switch occurs at time t_(j). That is

$\begin{matrix}{{y(t)} = \{ \begin{matrix}{- 1} & {t_{0} \leq t < t_{1}} & {{S(t)} = 0} \\1 & {t_{1} \leq t < t_{2}} & {{S(t)} = 1} \\{- 1} & {t_{2} \leq t < t_{3}} & {{S(t)} = 2} \\1 & {t_{3} \leq t < t_{4}} & {{S(t)} = 3} \\\ldots & \ldots & \;\end{matrix} } & (113)\end{matrix}$

Then we may writeU _(c)(t)={circumflex over (σ)}_(x) ^((y(t)+1)/2)  (114)or alternatively

$\begin{matrix}{{U_{c}(t)} = {{\frac{1 - {y(t)}}{2}\overset{\hat{}}{\mathbb{I}}} + {\frac{1 + {y(t)}}{2}{\overset{\hat{}}{\sigma}}_{x}}}} & (115) \\{= {{\frac{y(t)}{2}( {{\overset{\hat{}}{\sigma}}_{x} - \overset{\hat{}}{\mathbb{I}}} )} + {\frac{1}{2}{( {{\overset{\hat{}}{\sigma}}_{x} + \overset{\hat{}}{\mathbb{I}}} ).}}}} & (116)\end{matrix}$

Now assuming dephasingH ₀(t)=β_(z)(t){circumflex over (σ)}_(z)  (117)

the noise operator therefore takes the formA ^((x))={circumflex over (σ)}_(z)  (118)so that the toggling frame noise operator is given by

$\begin{matrix}{{{\overset{\sim}{A}}_{0}^{(z)}(t)} = {{U_{c}^{\dagger}(t)}{A^{(z)}(t)}{U_{c}(t)}}} & (119) \\{= {( {{\frac{1 - {y(t)}}{2}\overset{\hat{}}{\mathbb{I}}} + {\frac{1 + {y(t)}}{2}{\overset{\hat{}}{\sigma}}_{x}}} ){{\overset{\hat{}}{\sigma}}_{z}( {{\frac{1 - {y(t)}}{2}\overset{\hat{}}{\mathbb{I}}} + {\frac{1 + {y(t)}}{2}{\overset{\hat{}}{\sigma}}_{x}}} )}}} & (120) \\{= \begin{bmatrix}{- {y(t)}} & 0 \\0 & {y(t)}\end{bmatrix}} & (121)\end{matrix}$after expanding. Therefore in the Fourier domain we obtain

$\begin{matrix}{{\mathcal{F}\lbrack {{\overset{\sim}{A}}_{0}^{(z)}(t)} \rbrack} = \begin{bmatrix}{- {Y(\omega)}} & 0 \\0 & {Y(\omega)}\end{bmatrix}} & (122)\end{matrix}$defining the Fourier transform of the switching functionY(ω)=

[y(t)]  (123)

The dephasing filter function is therefore given by

$\begin{matrix}{{F_{z}(\omega)} = {\frac{1}{2}{\sum\limits_{l = 1}^{2}{\sum\limits_{q = 1}^{2}{\lbrack {\mathcal{F}\lbrack {{\overset{\sim}{A}}_{0}^{(i)}(t)} \rbrack} \rbrack_{lq}}^{2}}}}} & (124) \\{= {\frac{1}{2}( {{{- {Y(\omega)}}}^{2} + {{Y(\omega)}}^{2}} )}} & (125) \\{= {{Y(\omega)}}^{2}} & (126)\end{matrix}$

NOTE: this result is in fact independent of the sign convention of theswitching function due to the fact that we are taking the modulus squarein the end. That is, |

[y(t)]|²=|−

[y(t)]²=|

[y(t)]|²=Y(ω). Finally, we may write

$\begin{matrix}{{Y(\omega)} = {\sum\limits_{j = 0}^{n}{\int_{t_{j}}^{t_{j + 1}}{{y(t)}e^{{- i}\;\omega\; t}{dt}}}}} & (127) \\{= {\sum\limits_{j = 0}^{n}{\int_{t_{j}}^{t_{j + 1}}{( {- 1} )^{j}e^{{- i}\;\omega\; t}{dt}}}}} & (128) \\{= {\sum\limits_{j = 0}^{n}{( {- 1} )^{j}{\frac{1}{{- i}\;\omega}\lbrack {e^{{- i}\;\omega\; t_{j + 1}} - e^{{- i}\;\omega\; t_{j}}} \rbrack}}}} & (129) \\{= {\sum\limits_{j = 0}^{n}{( {- 1} )^{j}{\frac{1}{i\;\omega}\lbrack {e^{{- i}\;\delta_{j}{\omega\tau}} - e^{{- i}\;\delta_{j}\;\omega\;\tau}} \rbrack}}}} & (130)\end{matrix}$

Yielding

$\begin{matrix}{{F_{z}(\omega)} = {\frac{1}{\omega^{2}}{{\sum\limits_{j = 0}^{n}{( {- 1} )^{j}\lbrack {e^{{- i}\;\delta_{j}\omega\tau} - e^{{- i}\;\delta_{j + 1}\omega\tau}} \rbrack}}}^{2}}} & (131)\end{matrix}$For example:

For a spin echo sequence, set

${n = 1},{{{and}\mspace{14mu}( {\delta_{0},\delta_{1},\delta_{2}} )} = ( {0,\frac{1}{2},1} )},$then

$\begin{matrix}{{F_{z}(\omega)} = {\frac{1}{\omega^{2}}{{\lbrack {e^{{- i}\;\delta_{0}\omega\tau} - e^{{- i}\;\delta_{1}\omega\tau}} \rbrack - \lbrack {e^{{- i}\;\delta_{1}\omega\tau} - e^{{- i}\;\delta_{2}\omega\tau}} \rbrack}}^{2}}} & (132) \\{= {\frac{1}{\omega^{2}}{{\lbrack {1 - e^{{- i}\;{{\omega\tau}/2}}} \rbrack - \lbrack {e^{{- i}\;{{\omega\tau}/2}} - e^{{- i}\;{\omega\tau}}} \rbrack}}^{2}}} & (133) \\{= {\frac{1}{\omega^{2}}{\lbrack {1 - {2e^{{- i}\;{{\omega\tau}/2}}} + e^{{- i}\;{\omega\tau}}} \rbrack }^{2}}} & (134) \\{= {16{{\sin^{4}\lbrack \frac{\omega\tau}{4} \rbrack}/\omega^{2}}}} & (135)\end{matrix}$

Two-Qubit Gate Examples

The following description sets out some results for filter functionanalysis for two-qubit gates using the formalism set out above.

Representation

We define the following single- and two-qubit operator conventions asfollows

Single qubit

$\begin{matrix}{{ 1 \rangle = \begin{bmatrix}1 \\0\end{bmatrix}},{ 0 \rangle = \begin{bmatrix}0 \\1\end{bmatrix}}} & (201) \\{{{\overset{\hat{}}{\mathbb{I}}}_{1} = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}},{{\overset{\hat{}}{\sigma}}_{x} = \begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}},{{\overset{\hat{}}{\sigma}}_{y} = \begin{bmatrix}0 & {- i} \\i & 0\end{bmatrix}},{{\overset{\hat{}}{\sigma}}_{z} = \begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}} & (202)\end{matrix}$with commutation relations

$\begin{matrix}{\lbrack {{\hat{\sigma}}_{x},{\hat{\sigma}}_{y}} \rbrack = {2i{\hat{\sigma}}_{z}}} & (203) \\{\lbrack {{\hat{\sigma}}_{y},{\hat{\sigma}}_{z}} \rbrack = {2i{\hat{\sigma}}_{x}}} & (204) \\{\lbrack {{\hat{\sigma}}_{z},{\hat{\sigma}}_{x}} \rbrack = {2i{\hat{\sigma}}_{y}}} & (205) \\{\lbrack {{\hat{\sigma}}_{y},{\hat{\sigma}}_{x}} \rbrack = {{- 2}i{\hat{\sigma}}_{z}}} & (206) \\{\lbrack {{\hat{\sigma}}_{z},{\hat{\sigma}}_{Y}} \rbrack = {{- 2}i{\hat{\sigma}}_{x}}} & (207) \\{\lbrack {{\hat{\sigma}}_{x},{\hat{\sigma}}_{z}} \rbrack = {{- 2}i{\hat{\sigma}}_{y}}} & (208)\end{matrix}$

Two qubit

With the definitions provided below with reference to tensor products wewrite the two-qubit states

$\begin{matrix}{{ 11 \rangle \equiv { 1 \rangle \otimes  1 \rangle}} = \begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}} & (209) \\{{ 10 \rangle \equiv { 1 \rangle \otimes  0 \rangle}} = \begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}} & (210) \\{{ 01 \rangle \equiv { 0 \rangle \otimes  1 \rangle}} = \begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}} & (211) \\{{ 00 \rangle \equiv { 0 \rangle \otimes  0 \rangle}} = \begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} & (212)\end{matrix}$and operators

$\begin{matrix}{{{\overset{\hat{}}{\mathbb{I}}}_{2} \equiv {{\overset{\hat{}}{\mathbb{I}}}_{1} \otimes {\overset{\hat{}}{\mathbb{I}}}_{1}}} = {\begin{bmatrix}{\overset{\hat{}}{\mathbb{I}}}_{1} & 0 \\0 & {\overset{\hat{}}{\mathbb{I}}}_{1}\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} & (213) \\{{{\hat{I}X} \equiv {{\overset{\hat{}}{\mathbb{I}}}_{1} \otimes {\overset{\hat{}}{\sigma}}_{x}}} = {\begin{bmatrix}{\overset{\hat{}}{\sigma}}_{x} & 0 \\0 & {\overset{\hat{}}{\sigma}}_{x}\end{bmatrix} = \begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}}} & (214) \\{{{\hat{I}Y} \equiv {{\overset{\hat{}}{\mathbb{I}}}_{1} \otimes {\overset{\hat{}}{\sigma}}_{y}}} = {\begin{bmatrix}{\overset{\hat{}}{\sigma}}_{y} & 0 \\0 & {\overset{\hat{}}{\sigma}}_{y}\end{bmatrix} = \begin{bmatrix}0 & {- i} & 0 & 0 \\i & 0 & 0 & 0 \\0 & 0 & 0 & {- i} \\0 & 0 & i & 0\end{bmatrix}}} & (215) \\\vdots & (216) \\{{{\hat{X}X} \equiv {{\overset{\hat{}}{\sigma}}_{x} \otimes {\overset{\hat{}}{\sigma}}_{x}}} = {\begin{bmatrix}0 & {\overset{\hat{}}{\sigma}}_{x} \\{\overset{\hat{}}{\sigma}}_{x} & 0\end{bmatrix} = \begin{bmatrix}0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0\end{bmatrix}}} & (217) \\\vdots & (218) \\{{{\hat{X}Z} \equiv {{\overset{\hat{}}{\sigma}}_{z} \otimes {\overset{\hat{}}{\sigma}}_{y}}} = {\begin{bmatrix}0 & {\overset{\hat{}}{\sigma}}_{z} \\{\overset{\hat{}}{\sigma}}_{z} & 0\end{bmatrix} = \begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & {- 1} \\1 & 0 & 0 & 0 \\0 & {- 1} & 0 & 0\end{bmatrix}}} & (219) \\\vdots & (220) \\{{{\hat{Z}Y} \equiv {{\overset{\hat{}}{\sigma}}_{z} \otimes {\overset{\hat{}}{\sigma}}_{y}}} = {\begin{bmatrix}{\overset{\hat{}}{\sigma}}_{y} & 0 \\0 & {- {\hat{\sigma}}_{y}}\end{bmatrix} = \begin{bmatrix}0 & {- i} & 0 & 0 \\i & 0 & 0 & 0 \\0 & 0 & 0 & i \\0 & 0 & {- i} & 0\end{bmatrix}}} & (221) \\{{{\hat{Z}Z} \equiv {{\overset{\hat{}}{\sigma}}_{z} \otimes {\overset{\hat{}}{\sigma}}_{z}}} = {\begin{bmatrix}{\overset{\hat{}}{\sigma}}_{z} & 0 \\0 & {- {\hat{\sigma}}_{z}}\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {- 1} & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} & (222)\end{matrix}$

Two-qubit commutators

The commutators for all two-qubit operators is shown in the matrix inEq. (224). The margins on the left and top display all 16 two-qubitoperators, indexed from 1 to 16 down the rows and across the columns.The (i,j)th matrix element is defined as the commutator of the ithlisted operator (left margin) with the jth listed operator (top margin).For example, index 16 corresponds to ZZ and index 2 corresponds ÎX.Matrix element (16,2) displays 2i{circumflex over (Z)}Y (the prefactorcomes from multiplying the entire matrix by 2i). That is,

$\begin{matrix}{\lbrack {{\hat{Z}Z},{\hat{I}X}} \rbrack = {2i\hat{Z}Y}} & (223) \\{\begin{matrix}\overset{.}{II} \\\overset{.}{IX} \\\overset{.}{IY} \\\overset{.}{IZ} \\\overset{.}{XI} \\\overset{.}{XX} \\\overset{.}{XY} \\\overset{.}{XZ} \\\overset{.}{YI} \\\overset{.}{YX} \\\overset{.}{YY} \\\overset{.}{YZ} \\\overset{.}{ZI} \\\overset{.}{ZX} \\\overset{.}{ZY} \\\overset{.}{ZZ}\end{matrix}\overset{\begin{matrix}\overset{.}{II} & \overset{.}{IX} & \overset{.}{IY} & \overset{.}{IZ} & \overset{.}{XI} & \overset{.}{XX} & \overset{.}{XY} & \overset{.}{XZ} & \overset{.}{YI} & \overset{.}{YX} & \overset{.}{YY} & \overset{.}{YZ} & \overset{.}{ZI} & \overset{.}{ZX} & \overset{.}{\mspace{20mu}{ZY}} & \overset{.}{\mspace{20mu}{ZZ}}\end{matrix}}{\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \overset{.}{IZ} & {- \overset{.}{IY}} & 0 & 0 & \overset{.}{XZ} & {- \overset{.}{XY}} & 0 & 0 & \overset{.}{YZ} & {- \overset{.}{YY}} & 0 & 0 & \overset{.}{ZZ} & {- \overset{.}{ZY}} \\0 & {- \overset{.}{IZ}} & 0 & \overset{.}{IX} & 0 & {- \overset{.}{XZ}} & 0 & \overset{.}{XX} & 0 & {- \overset{.}{YZ}} & 0 & \overset{.}{YX} & 0 & {- \overset{.}{ZZ}} & 0 & \overset{.}{ZX} \\0 & \overset{.}{IY} & {- \overset{.}{IX}} & 0 & 0 & \overset{.}{XY} & {- \overset{.}{XX}} & 0 & 0 & \overset{.}{YY} & {- \overset{.}{YX}} & 0 & 0 & \overset{.}{ZY} & {- \overset{.}{ZX}} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \overset{.}{ZI} & \overset{.}{ZX} & \overset{.}{ZY} & \overset{.}{ZZ} & {- \overset{.}{YI}} & {- \overset{.}{YX}} & {- \overset{.}{YY}} & {- \overset{.}{YZ}} \\0 & 0 & \overset{.}{XZ} & {- \overset{.}{XY}} & 0 & 0 & \overset{.}{IZ} & {- \overset{.}{IY}} & \overset{.}{ZX} & \overset{.}{ZI} & 0 & 0 & {- \overset{.}{YX}} & {- \overset{.}{YI}} & 0 & 0 \\0 & {- \overset{.}{XZ}} & 0 & \overset{.}{XX} & 0 & {- \overset{.}{IZ}} & 0 & \overset{.}{IX} & \overset{.}{ZY} & 0 & \overset{.}{ZI} & 0 & {- \overset{.}{YY}} & 0 & {- \overset{.}{YI}} & 0 \\0 & \overset{.}{XY} & {- \overset{.}{XX}} & 0 & 0 & \overset{.}{IY} & {- \overset{.}{IX}} & 0 & \overset{.}{ZZ} & 0 & 0 & \overset{.}{ZI} & {- \overset{.}{YZ}} & 0 & 0 & {- \overset{.}{YI}} \\0 & 0 & 0 & 0 & {- \overset{.}{ZI}} & {- \overset{.}{ZX}} & {- \overset{.}{ZY}} & {- \overset{.}{ZZ}} & 0 & 0 & 0 & 0 & \overset{.}{XI} & \overset{.}{XX} & \overset{.}{XY} & \overset{.}{XZ} \\0 & 0 & \overset{.}{YZ} & {- \overset{.}{YY}} & {- \overset{.}{ZX}} & {- \overset{.}{ZI}} & 0 & 0 & 0 & 0 & \overset{.}{IZ} & {- \overset{.}{IY}} & \overset{.}{XX} & \overset{.}{XI} & 0 & 0 \\0 & {- \overset{.}{YZ}} & 0 & \overset{.}{YX} & {- \overset{.}{ZY}} & 0 & {- \overset{.}{ZI}} & 0 & 0 & {- \overset{.}{IZ}} & 0 & \overset{.}{IX} & \overset{.}{XY} & 0 & \overset{.}{XI} & 0 \\0 & \overset{.}{YY} & {- \overset{.}{YX}} & 0 & {- \overset{.}{ZZ}} & 0 & 0 & {- \overset{.}{ZI}} & 0 & \overset{.}{IY} & {- \overset{.}{IX}} & 0 & \overset{.}{XZ} & 0 & 0 & \overset{.}{XI} \\0 & 0 & 0 & 0 & \overset{.}{YI} & \overset{.}{YX} & \overset{.}{YY} & \overset{.}{YZ} & {- \overset{.}{XI}} & {- \overset{.}{XX}} & {- \overset{.}{XY}} & {- \overset{.}{XZ}} & 0 & 0 & 0 & 0 \\0 & 0 & \overset{.}{ZZ} & {- \overset{.}{ZY}} & \overset{.}{YX} & \overset{.}{YI} & 0 & 0 & {- \overset{.}{XX}} & {- \overset{.}{XI}} & 0 & 0 & 0 & 0 & \overset{.}{IZ} & {- \overset{.}{IY}} \\0 & {- \overset{.}{ZZ}} & 0 & \overset{.}{ZX} & \overset{.}{YY} & 0 & \overset{.}{YI} & 0 & {- \overset{.}{XY}} & 0 & {- \overset{.}{XI}} & 0 & 0 & {- \overset{.}{IZ}} & 0 & {IX} \\0 & \overset{.}{ZY} & {- \overset{.}{ZX}} & 0 & \overset{.}{YZ} & 0 & 0 & \overset{.}{YI} & {- \overset{.}{XZ}} & 0 & 0 & {- \overset{.}{XI}} & 0 & \overset{.}{IY} & {- \overset{.}{IX}} & 0\end{bmatrix}} \times 2i} & (2.24)\end{matrix}$These commutators may be derived using Eq. (406). For example,

$\begin{matrix}{\lbrack {{\overset{\hat{}}{Z}Z},{\overset{\hat{}}{I}X}} \rbrack = {{( {\overset{\hat{}}{Z}Z} )( {\overset{\hat{}}{I}X} )} - {( {\overset{\hat{}}{I}X} )( {\overset{\hat{}}{Z}Z} )}}} & (225) \\{= {{( {{\overset{\hat{}}{\sigma}}_{z} \otimes {\overset{\hat{}}{\sigma}}_{z}} )( {{\overset{\hat{}}{\mathbb{I}}}_{2} \otimes {\overset{\hat{}}{\sigma}}_{x}} )} - {( {{\overset{\hat{}}{\mathbb{I}}}_{2} \otimes {\overset{\hat{}}{\sigma}}_{x}} )( {{\overset{\hat{}}{\sigma}}_{z} \otimes {\overset{\hat{}}{\sigma}}_{z}} )}}} & (226) \\{= {{( {{\overset{\hat{}}{\sigma}}_{z}{\overset{\hat{}}{\mathbb{I}}}_{2}} ) \otimes ( {{\overset{\hat{}}{\sigma}}_{z}{\overset{\hat{}}{\sigma}}_{x}} )} - {( {{\overset{\hat{}}{\mathbb{I}}}_{2}{\overset{\hat{}}{\sigma}}_{z}} ) \otimes ( {{\overset{\hat{}}{\sigma}}_{x}{\overset{\hat{}}{\sigma}}_{z}} )}}} & (227) \\{= {{{\overset{\hat{}}{\sigma}}_{z} \otimes ( {i{\overset{\hat{}}{\sigma}}_{y}} )} - {( {\overset{\hat{}}{\sigma}}_{z} ) \otimes ( {{- i}{\overset{\hat{}}{\sigma}}_{y}} )}}} & (228) \\{= {{i\overset{\hat{}}{Z}Y} + {i\overset{\hat{}}{Z}Y}}} & (229) \\{= {2i\overset{\hat{}}{Z}Y}} & (230)\end{matrix}$

We are interested in examining subspaces of the 16-element operatorspace possessing commutator relations sharing the structure of thecommutator relationships of the single-qubit Pauli operators. Working inthese subspaces provides a path towards mapping familiar single-qubitcontrols to two-qubit systems. Below consider two such subspaces ofinterest, due to their including the two-qubit interaction terms{circumflex over (Z)}Z and {circumflex over (X)}X which may beengineered in physical systems.

Effective compensating pulses: {circumflex over (Z)}{circumflex over(Z)} subspace

Consider the two-qubit control HamiltonianH _(c)=α_(zz)(t){circumflex over (Z)}Z+α _(ix)(t)ÎX+α_(zy)(t){circumflex over (Z)}Y  (231)mapping to the effective single qubit Hamiltonian

$\begin{matrix}{H_{c} = {\frac{1}{2}( {{{\Omega_{x}(t)}{\hat{S}}_{x}} + {{\Omega_{y}(t)}{\hat{S}}_{y}} + {{\Omega_{z}(t)}{\hat{S}}_{z}}} )}} & (232)\end{matrix}$where the effective operators are defined asŜ _(x) ={circumflex over (Z)}Z, Ŝ _(y) =ÎX, Ŝ _(z) ={circumflex over(Z)}Y  (233)and the control amplitudes are defined as

$\begin{matrix}{{\alpha_{zz} = \frac{\Omega_{x}}{2}},{\alpha_{ix} = \frac{\Omega_{y}}{2}},{\alpha_{zy} = \frac{\Omega_{z}}{2}}} & (234)\end{matrix}$

Then using Eq. (406) or reading of the table in Eq. (224), we obtain thefollowing commutation relations

$\begin{matrix}{\lbrack {{\hat{S}}_{x},{\hat{S}}_{y}} \rbrack = {2i{\hat{S}}_{z}}} & (235) \\{\lbrack {{\hat{S}}_{y},{\hat{S}}_{z}} \rbrack = {2i{\hat{S}}_{x}}} & (236) \\{\lbrack {{\hat{S}}_{z},{\hat{S}}_{x}} \rbrack = {2i{\hat{S}}_{y}}} & (237) \\{\lbrack {{\hat{S}}_{y},{\hat{S}}_{x}} \rbrack = {{- 2}i{\hat{S}}_{z}}} & (238) \\{\lbrack {{\hat{S}}_{z},{\hat{S}}_{y}} \rbrack = {{- 2}i{\hat{S}}_{x}}} & (239) \\{\lbrack {{\hat{S}}_{x},{\hat{S}}_{z}} \rbrack = {{- 2}i{\hat{S}}_{y}}} & (240)\end{matrix}$

Since the Ŝ_(i) satisfy the commutation relations for Pauli operators,Eq. (232) is equivalent to a single-qubit Hamiltonian, so we mayimplement any of the control techniques known to work in that space.That is, any controls known to have compensating properties in thesingle-qubit basis {{circumflex over (σ)}_(x), {circumflex over(σ)}_(y), {circumflex over (σ)}_(z)} should map to two-qubit systems,with controls (and noise) spanned by this basis {Ŝ_(x),Ŝ_(y),Ŝ_(z)}. Weconsider two cases below, and show the corresponding filter functions,showing agreement with expected performance.

Effective primitive π_(x)

For an interaction time τ, set

$\begin{matrix}{{\Omega_{x} = \frac{\pi}{\tau}},{\Omega_{y} = 0},{\Omega_{z} = 0.}} & (241)\end{matrix}$

This resembles a primitive π pulse around the single-qubit Ŝ_(x) axis,but in the two-qubit space maps to accumulating a π phase on the{circumflex over (Z)}Z interaction term. One expects filter function forthis operation in the two-qubit space to inherit the same properties(i.e. filter order) as the primitive single-qubit π_(x) pulse.

Effective BB1-protected π_(x) in multiqubit subspace for ZZ interactions

In one example, we can achieve noise-filtering multiqubit logicoperations by crafting controls inspired by single-qubit techniques, butnow operating in the noise filtering framework described in thisdisclosure for multiqubit operations. We can now implement the 4-segmentBB1 pulse to achieve the same effective operation on Ŝ_(x)≡{circumflexover (Z)}Z with augmented noise-filtering properties. Let {circumflexover (R)}(θ,ϕ) denote a rotation of the Bloch vector about the Ŝ_(x)cos(ϕ)+Ŝ_(y) sin(ϕ) axis, through an angle θ. Then the BB1 pulseimplementing a net θ rotation about Ŝ_(x) is written (operators appliedfrom right to left)BB1(θ)={circumflex over (R)}(π,ϕ_(BB1)){circumflex over(R)}(2π,3ϕ_(BB1)){circumflex over (R)}(π,ϕ_(BB1)){circumflex over(R)}(π,0)  (242)where

$\begin{matrix}{\phi_{{BB}\; 1} = {\cos^{- 1}( {- \frac{\theta}{4\pi}} )}} & (243)\end{matrix}$implemented with constant Rabi rate Ω₀ over a series of segments withdurations τ_(l), l∈{1, . . . , 4}, this is written

$\begin{matrix}{{{BB}\; 1(\theta)} = {\begin{matrix}P_{1} \\\; \\P_{2} \\\; \\P_{3} \\\; \\P_{4}\end{matrix}{\overset{\begin{matrix}\; & {\mspace{25mu}\Omega_{x}} & \; & \; & \; & \; & {\mspace{14mu}\Omega_{y}} & \mspace{20mu} & {\mspace{20mu}\Omega_{z}} & {\tau_{1\;}\mspace{11mu}}\end{matrix}}{\begin{bmatrix}{\Omega_{0}{\cos(0)}} & {\Omega_{0}{\sin(0)}} & 0 & \frac{\theta}{\Omega_{0}} \\{\Omega_{0}{\cos( \phi_{{BB}\; 1} )}} & {\Omega_{0}{\sin( \phi_{{BB}\; 1} )}} & 0 & \frac{\pi}{\Omega_{0}} \\{\Omega_{0}{\cos( {3\phi_{{BB}\; 1}} )}} & {\Omega_{0}{\sin( {3\phi_{{BB}\; 1}} )}} & 0 & \frac{2\pi}{\Omega_{0}} \\{\Omega_{0}{\cos( \phi_{{BB}\; 1} )}} & {\Omega_{0}{\sin( \phi_{{BB}\; 1} )}} & 0 & \frac{\pi}{\Omega_{0}}\end{bmatrix}}.}}} & (244)\end{matrix}$

FIG. 5 illustrates filter functions for a two-qubit system, comparingeffective primitive π_(x) and effective BB1(π) implemented in the{circumflex over (Z)}Z subspace. BB1 exhibits higher filter order thanprimitive.

The BB1(π) pulse achieves the same net operation as the primitive πpulse, but compensates for errors in the amplitude (Rabi rate). That is,the amplitude filter function for BB1(π) should exhibit higher filterorder than the primitive, for a noise operator proportional to thecontrol. Here amplitude noise means common Ŝ_(x), Ŝ_(y) noise. In thetwo-qubit picture this implies common noise on controls {circumflex over(Z)}Z and ÎX. FIG. 5 demonstrates this result. It is in general achallenge to calculate this filtering behavior, absent the disclosedmethod.

Effective compensating pulses: {circumflex over (X)}{circumflex over(X)} interactions

Consider the two-qubit control HamiltonianH _(c)=α_(xx)(t){circumflex over (X)}X+α _(iy)(t)ÎY+α_(xz)(t){circumflex over (X)}Z  (245)mapping to the effective single qubit Hamiltonian

$\begin{matrix}{H_{c} = {\frac{1}{2}( {{{\Omega_{x}(t)}{\hat{S}}_{x}} + {{\Omega_{y}(t)}{\hat{S}}_{y}} + {{\Omega_{z}(t)}{\hat{S}}_{z}}} )}} & (246)\end{matrix}$where the effective operators Ŝ_(i) are defined asŜ _(x) ={circumflex over (X)}X, Ŝ _(y) =ÎY, Ŝ _(z) ={circumflex over(X)}Z  (247)and the control amplitudes are defined as

$\begin{matrix}{{\alpha_{xx} = \frac{\Omega_{x}}{2}},{\alpha_{iy} = \frac{\Omega_{y}}{2}},{\alpha_{xz} = \frac{\Omega_{z}}{2}}} & (248)\end{matrix}$

Then using Eq. (406) below or reading of the table in Eq. (224), weobtain the following commutation relations

$\begin{matrix}{\lbrack {{\hat{S}}_{x},{\hat{S}}_{y}} \rbrack = {2i{\hat{S}}_{z}}} & (249) \\{\lbrack {{\hat{S}}_{y},{\hat{S}}_{z}} \rbrack = {2i{\hat{S}}_{x}}} & (250) \\{\lbrack {{\hat{S}}_{z},{\hat{S}}_{x}} \rbrack = {2i{\hat{S}}_{y}}} & (251) \\{\lbrack {{\hat{S}}_{y},{\hat{S}}_{z}} \rbrack = {{- 2}i{\hat{S}}_{x}}} & (252) \\{\lbrack {{\hat{S}}_{z},{\hat{S}}_{y}} \rbrack = {{- 2}i{\hat{S}}_{x}}} & (253) \\{\lbrack {{\hat{S}}_{x},{\hat{S}}_{z}} \rbrack = {{- 2}i{\hat{S}}_{y}}} & (254)\end{matrix}$

Since the Ŝ_(i) satisfy the commutation relations for Pauli operators,Eq. (246) is equivalent to a single-qubit Hamiltonian, so we mayimplement any of the control techniques known to work in that space.That is, any controls known to have compensating properties in thesingle-qubit basis {{circumflex over (σ)}_(x),{circumflex over(σ)}_(y),{circumflex over (σ)}_(z)} should map to two-qubit systems,with controls (and noise) spanned by this basis {Ŝ_(x),Ŝ_(y),Ŝ_(z),}. Weconsider two cases below, and show the corresponding filter functions,showing agreement with expected performance. Again, we consider twocases below, and show the corresponding filter functions, showingagreement with expected performance.

Effective primitive π_(x)

For an interaction time τ, set

$\begin{matrix}{{{\Omega_{x} = \frac{\pi}{\tau}},{\Omega_{y} = 0},{\Omega_{z} = 0}}.} & (255)\end{matrix}$

This resembles a primitive π pulse around the single-qubit Ŝ_(x) axis,but in the two-qubit space maps to accumulating a π phase on the{circumflex over (X)}X interaction term. One expects filter function forthis operation in the two-qubit space to inherit the same properties(i.e. filter order) as the primitive single-qubit π, pulse.

Effective BB1-protected π_(x) in multiqubit subspace for XX interactions

In another example we can also implement the 4-segment BB1 pulse toachieve the same effective operation on Ŝ_(x)≡{circumflex over (X)}Xwith augmented noise filtering properties. Let {circumflex over (R)}(θ,ϕ) denote a rotation of the Bloch vector about the Ŝ_(x) cos(ϕ)+Ŝ_(y)sin (ϕ) axis, through an angle θ. Then the BB1 pulse implementing a netθ rotation about Ŝ_(x) is written (operators applied from right to left)BB1(θ)={circumflex over (R)}(π,ϕ_(BB1)){circumflex over(R)}(2π,3ϕ_(BB1)){circumflex over (R)}(π,ϕ_(BB1)){circumflex over(R)}(π,0)  (256)where

$\begin{matrix}{\phi_{{BB}\; 1} = {\cos^{- 1}( {- \frac{\theta}{4\pi}} )}} & (257)\end{matrix}$implemented with constant Rabi rate Ω₀ over as a series of segments withdurations τ_(l), l∈{1, . . . , 4}, this is written

$\begin{matrix}{{{BB}\; 1(\theta)} = {\begin{matrix}P_{1} \\\; \\P_{2} \\\; \\P_{3} \\\; \\P_{4}\end{matrix}{\overset{\begin{matrix}\; & {\mspace{25mu}\Omega_{x}} & \; & \; & \; & \; & {\mspace{14mu}\Omega_{y}} & \mspace{20mu} & {\mspace{20mu}\Omega_{z}} & \tau_{1\mspace{11mu}}\end{matrix}}{\begin{bmatrix}{\Omega_{0}{\cos(0)}} & {\Omega_{0}{\sin(0)}} & 0 & \frac{\theta}{\Omega_{0}} \\{\Omega_{0}{\cos( \phi_{{BB}\; 1} )}} & {\Omega_{0}{\sin( \phi_{{BB}\; 1} )}} & 0 & \frac{\pi}{\Omega_{0}} \\{\Omega_{0}{\cos( {3\phi_{{BB}\; 1}} )}} & {\Omega_{0}{\sin( {3\phi_{{BB}\; 1}} )}} & 0 & \frac{2\pi}{\Omega_{0}} \\{\Omega_{0}{\cos( \phi_{{BB}\; 1} )}} & {\Omega_{0}{\sin( \phi_{{BB}\; 1} )}} & 0 & \frac{\pi}{\Omega_{0}}\end{bmatrix}}.}}} & (258)\end{matrix}$

FIG. 6 illustrates filter functions for a two-qubit system, comparingeffective primitive π_(x) and effective BB1(π) implemented in the{circumflex over (X)}X subspace. BB1 exhibits higher filter order thanprimitive.

The BB1(π) pulse achieves the same net operation as the primitive πpulse, but compensates for errors in the amplitude (Rabi rate). That is,the amplitude filter function for BB1(π) should exhibit higher filterorder than the primitive, for a noise operator proportional to thecontrol. Here amplitude noise means common Ŝ_(x),Ŝ_(y) noise. In thetwo-qubit picture this implies common noise on controls {circumflex over(X)}X and ÎY. FIG. 6 demonstrates this result, as calculated using thefilter functions which are the subject of this disclosure.

Mathematical Preliminaries

The following description provides mathematical formulations used above.

Tensor product

Definition 1.1 If A is a m×n matrix and B is a p×q matrix, define theKronecker product A⊗B as

$\begin{matrix}{{A \otimes B} = \begin{Bmatrix}{a_{1,1}B} & \ldots & {a_{1,n}B} \\\vdots & \ddots & \vdots \\{a_{m,1}B} & \ldots & {a_{m,n}B}\end{Bmatrix}} & (401)\end{matrix}$yielding the mp×nq matrix.

Then the following properties hold

Property 1.1 Define matricesA(m×n)  (402)B(p×q)  (403)C(n×r)  (404)D(q×s)  (405)then(A⊗B)(C⊗D)=(AC)⊗(BD)  (406)

In particular

Property 1.2 Let {right arrow over (u)} be a m×1 column vector and{right arrow over (v)} be p×1 column vector, then the Kronecker productresults in the (mp×1) column vector

$\begin{matrix}{{\overset{->}{u} \otimes \overset{->}{v}} = {\begin{bmatrix}{u_{1}\overset{->}{v}} \\\vdots \\{u_{m}\overset{->}{v}}\end{bmatrix} = \begin{bmatrix}{u_{1}v_{1}} \\{u_{1}v_{2}} \\\vdots \\{u_{m}v_{p­1}} \\{u_{m}v_{p}}\end{bmatrix}}} & (407) \\{{( {A \otimes B} )( {\overset{->}{u} \otimes \overset{->}{v}} )} = {( {A\overset{->}{u}} ) \otimes ( {B\overset{->}{v}} )}} & (408)\end{matrix}$

Rephrasing using quantum mechanical kets and operators,

Property 1.3 Let

_(m) and

_(p) be m- and p-dimensional Hilbert spaces, on which we defineoperatorsA∈

_(m) , B∈

_(p)  (409)and kets|u>,Aβu>∈

_(m) , |v>,B|v>∈

_(p)  (410)Then|uv>≡|u>⊗|v>∈

_(mp) , A⊗B∈

_(mp)  (411)are a ket and operator on the joint Hilbert space

_(mp)=

⊗

_(p) such that(A⊗B)(|u>⊗|v>)=A|u>⊗B|v>  (412)

We define the outer product

Definition 1.2 Let |u>,|v>∈

be kets on the same Hilbert space. Then the outer productuv≡(|u>)^(†) ⊗|v>  (413)defined an operator on

Then the following property holds

Property 1.4 Let |a>,|b>,|c>,|d>∈

be kets on the same Hilbert space and define the operatorsab,cd∈

  (414)

Then the operator ab⊗cd∈

⊗

defined on the joint Hilbert space satisfiesab⊗cd=acbd  (415)where|ac>=|a>⊗|b>∈

⊗

|bd>=|b>└|d>∈

⊗

  (416)

It will be appreciated by persons skilled in the art that numerousvariations and/or modifications may be made to the above-describedembodiments, without departing from the broad general scope of thepresent disclosure. The present embodiments are, therefore, to beconsidered in all respects as illustrative and not restrictive.

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The invention claimed is:
 1. A method for evaluating and improvingperformance of a control implementation on a quantum processorcomprising multiple qubits in the presence of noise, the methodcomprising: modelling the noise by decomposing noise interactionsdescribed by a multi-qubit noise Hamiltonian into multiple contributorynoise channels, each channel generating noise dynamics described by aunique noise-axis operator A{circumflex over ( )}{(i)}, where (i)indexes the ith noise channel; determining, for a given controlimplementation, the unique filter function for each noise channelrepresenting susceptibility of the multi-qubit system to the associatednoise dynamics, each of the multiple filter functions being based on afrequency transformation ($\mathcal{F}$) of the noise axis operator(A{circumflex over ( )}{(i)}) of the corresponding noise channel (i) tothereby evaluate the performance of the control implementation;determining an optimised control sequence based on the filter functionto reduce the susceptibility of the multi-qubit system to the noisechannels, thereby reducing the effective interaction with themulti-qubit noise Hamiltonian; and applying the optimised controlsequence to the multi-qubit system to control the quantum processor tothereby improve the performance of the control implementation.
 2. Themethod of claim 1, wherein the multi-qubit noise axis operators arelinear operators.
 3. The method of claim 1, wherein the combination ofthe noise axis operators comprises a weighting of each noise axisoperator by a random variable.
 4. The method of claim 1, wherein thecombination is a linear combination of the noise axis operators weightedby respective random variables.
 5. The method of claim 1, wherein eachof the multiple filter functions is based on a frequency transformationof a control Hamiltonian (U_c) applied to the noise axis operator. 6.The method of claim 5, wherein the control Hamiltonian represents anoperation on multiple qubits.
 7. The method of claim 6, wherein a gateset of the quantum processor comprises entangling operations and thecontrol Hamiltonian includes the entangling operations between themultiple qubits.
 8. The method of claim 5, wherein the controlHamiltonian (U_c) applied to the noise axis operator forms a togglingframe noise axis operator.
 9. The method of claim 8, wherein thetoggling frame noise axis operator represents control dynamics relativeto the noise axis excluding stochastic content of the noise.
 10. Themethod of claim 1, wherein the noise axis operator is in the Hilbertspace of the multiple qubits.
 11. The method of claim 1, wherein thenoise axis operator is time varying.
 12. The method of claim 1, whereinthe dimensions of the noise axis operator are equal to the dimensions ofa Hamiltonian of the multiple qubits.
 13. The method of claim 1, whereinthe noise axis operator is based on a non-Markovian error model.
 14. Themethod of claim 1, wherein the noise axis operator is based on one ormore measurements of an environment of the quantum processor.
 15. Themethod of claim 1, wherein each filter function is based on a sum overdimensions of the noise axis operator of the frequency transformation.16. The method of claim 1, wherein determining a control sequence basedon the filter function is to reduce the noise influence on the operationon the multiple qubits.
 17. The method of claim 1, further comprising:mapping operators for a multi-qubit system to an effective controlHamiltonian of reduced dimensionality, wherein each of the multiplefilter functions is based on a frequency transformation of the effectivecontrol Hamiltonian (U_c) applied to the noise axis operator such thatthe optimized control sequence represents a multi-qubit controlsolution.
 18. Software that, when executed by a computer, causes thecomputer to perform the method of claim
 1. 19. A quantum processorcomprising multiple qubits and a controller configured to: model thenoise by decomposing noise interactions described by a multi-qubit noiseHamiltonian into multiple contributory noise channels, each channelgenerating noise dynamics described by a unique noise-axis operatorA{circumflex over ( )}{(i)}, where (i) indexes the ith noise channel;determine, for a given control implementation, the unique filterfunction for each noise channel representing susceptibility of themulti-qubit system to the associated noise dynamics, each of themultiple filter functions being based on a frequency transformation($\mathcal{F}$) of the noise axis operator (A{circumflex over ( )}{(i)})of the corresponding noise channel (i) to thereby evaluate theperformance of the control implementation; determine an optimisedcontrol sequence based on the filter function to reduce thesusceptibility of the multi-qubit system to the noise channels, therebyreducing the effective interaction with the multi-qubit noiseHamiltonian; and apply the optimised control sequence to the multi-qubitsystem to control the quantum processor to thereby improve theperformance of the control implementation.